where \(\gamma _{p,v}\)  is the geodesic of \((M,g)\)  with the initial conditions \((p,v)\in SM\) . Since the boundary of \(M\)  is strictly convex \({\tau _{\text{exit}}}(p,v)\)  is the first time when the geodesic \(\gamma _{p,v}\)  hits the boundary and \((-{\tau _{\text{exit}}}(p,-v),{\tau _{\text{exit}}}(p,v))\)  is the maximal interval where the geodesic \(\gamma _{p,v}\)  is defined. We do not assume that \({\tau _{\text{exit}}}(p,v)<\infty \)  for all \((p,v)\in SM\) . That is, \((M,g)\)  may have trapped geodesics.
It is well known that the exit time function, on manifolds with strictly convex boundary, is continuous and its restriction on \(SM \setminus T\partial M\)  (for those \((p,v)\in SM\)  for which \({\tau _{\text{exit}}}(p,v)<\infty \) ) is smooth (See for instance [1]}).
